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Investigation of the dynamics of physical systems by supersymmetric quantum mechanics

Andrey M. Pupasov-Maksimov, B. F. Samsonov, Jean-Marc Sparenberg

Published Jun 3, 2010• 0 citations
OpenAlex

Abstract

Relations between propagators and Green functions of Hamiltonians which are SUSY partners have been obtained. New exact propagators for the family of multi-well, time-dependent and non-hermitian potentials have been calculated. Non-conservative SUSY transformation has been studied in the case of multichannel Schrodinger equation with different thresholds. Spectrum (bound/virtual states and resonances) of the non-conservative SUSY partner of zero potential has been founded. Exactly solvable model of the magnetic induced Feshbach resonance has been constructed. This model was tested in the case of Rb$^{85}$. Conservative SUSY transformations of the first and the second order have been studied in the case of multichannel Schrodinger equation with equal thresholds. Transformations which introduce non-trivial coupling between scattering channels have been founded. The first order SUSY transformation which preserves $S$-matrix eigen-phase shifts and modifies mixing parameter has been founded in the case of two channel scattering with partial waves of different parities. In the case of coinciding parities we have found the second order SUSY transformation which preserves $S$-matrix eigen-phase shifts and modifies mixing parameter. Phenomenological two channel $^3S_1-^3D_1$ neutron-proton potential has been obtained by using single channel, phase equivalent and coupling SUSY transformations applied to zero potential.

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Relations between propagators and Green functions of Hamiltonians which are SUSY partners have been obtained. New exact propagators for the family of multi-well, time-dependent and non-hermitian potentials have been calculated. Non-conservative SUSY transformation has been studied in the case of multichannel Schrodinger equation with different thresholds. Spectrum (bound/virtual states and resonances) of the non-conservative SUSY partner of zero potential has been founded. Exactly solvable model of the magnetic induced Feshbach resonance has been constructed. This model was tested in the case of Rb85^{85}85. Conservative SUSY transformations of the first and the second order have been studied in the case of multichannel Schrodinger equation with equal thresholds. Transformations which introduce non-trivial coupling between scattering channels have been founded. The first order SUSY transformation which preserves SSS-matrix eigen-phase shifts and modifies mixing parameter has been founded in the case of two channel scattering with partial waves of different parities. In the case of coinciding parities we have found the second order SUSY transformation which preserves SSS-matrix eigen-phase shifts and modifies mixing parameter. Phenomenological two channel 3S1−3D1^3S_1-^3D_13S1​−3D1​ neutron-proton potential has been obtained by using single channel, phase equivalent and coupling SUSY transformations applied to zero potential.
Supersymmetry (SUSY) postulates that for every fermion there is boson of equal mass (i.e. energy). In quantum mechanics this comes about because for every quantum Hamiltonian there is a partner Hamiltonian that has the same energy spectrum above the ground state of the original system. In particle physics, one “sector” is populated by bosons and the other sector by fermions and SUSY predicts that the lowest lying fermion state is energetically degenerate with the first excited boson state. Evidence for SUSY has proven to be elusive and it is now believed that SUSY is a broken symmetry. In January 2009 at a conference dedicated to Bob Wyatt, we began to look at SUSY as a way to develop new computational methods and approaches. Up until now, SUSY has been more of a mathematical technique that has been used more or less as a way to obtain stationary solutions to the Schrödinger equation for a variety of T&F Cat # K11224, Chapter 4, Page 54, 15-9-2010 one-dimensional potential systems. In this chapter, we will discuss some of the work we have been doing in developing “SUSY” inspired methods for performing quantum many-body calculations and quantum scattering calculations. We begin with a brief overview of the SUSY theory and some of its elementary results. We then discuss how we have used the approach to develop both analytical and numerical solutions of the stationary Schrödinger equation. Finally, we conclude by discussing our recent extension of SUSY to higher dimensions and for scattering theory.
Starting from a system of NNN radial Schr\"odinger equations with a vanishing potential and finite threshold differences between the channels, a coupled $N\ifmmode\times\else\texttimes\fi{}N$ exactly solvable potential model is obtained with the help of a single nonconservative supersymmetric transformation. The obtained potential matrix, which subsumes a result obtained in the literature, has a compact analytical form, as does its Jost matrix. It depends on N(N+1)/2N(N+1)/2N(N+1)/2 unconstrained parameters and on one upper-bounded parameter, the factorization energy. A detailed study of the model is done for the $2\ifmmode\times\else\texttimes\fi{}2$ case: a geometrical analysis of the zeros of the Jost-matrix determinant shows that the model has zero, one, or two bound states, and zero or one resonance; a compact formula for the open-channel scattering length is obtained. As a first physical application, exactly solvable $2\ifmmode\times\else\texttimes\fi{}2$ atom-atom interaction potentials are constructed, for cases where a magnetic Feshbach resonance interacts with a bound or virtual state close to threshold, which results in a large background scattering length.
This undergraduate thesis contains one dimensional introductory treatment of Supersymmetric Quantum Mechanics (SUSY QM). The factorisation method is used to obtain Partner Hamiltonians, and the notions of partner potentials and superpotential are introduced. It demonstrated how energy spectra and eigenfunctions of partner potentials can be correlated for the cases of unbroken and broken supersymmetry. Generalisations of these cases attributed to Sets of Hamiltonians and shape invariant potentials are introduced. The devel oped formalisation is used to obtain energy eigenfunctions and spectra of partner potentials of some well-known quantum systems, namely the Infinite Square Well and the Free Particle. In these systems, the partner potentials are trigonometric and hyperbolic functions, respectively. A comparison of these two physically distinct but mathematically related partner potentials is provided. A new philosophical interpretation of SUSY QM is introduced, with an ontological discussion of supersymmetric notions and SUSY QM. The current observation problem of SUSY is discussed, and on that basis, two opposing arguments are presented. The method of Falsification by Karl Popper is briefly introduced, and an argument is produced regarding its application to SUSY.
We demonstrate the supersymmetric (SUSY) structure of a generalized non-Hermitian driven two-level system. By applying a unitary rotation and decoupling the resulting differential equations, we map the time-dependent dynamics onto a SUSY pair of stationary-like Schrödinger equations. This framework allows us to derive exactly solvable, complex time-dependent driving functions from the spectral analysis of auxiliary stationary systems. We show that these complex couplings induce a variety of non-trivial qubit state transitions, offering precise control mechanisms in non-conservative regimes. Our results provide a versatile analytical tool for state engineering in platforms such as nuclear magnetic resonance and non-Hermitian optics with gain and loss profiles.
Abstract In this paper, we use some well-known techniques of Supersymmetric QuantumMechanics (SUSYQM) namely the factorization method and shape invariance, to generate new analytically solvable potentials from some interacting fermionic models in nonrelativistic limits. These systems are described by the ordinary and the harmonically trapped Schrödinger-Pauli particle models and the Dirac-Coulomb Hamiltonian, this latter being set in its nonrelativistic limits. The spectrum for each of these models is obtained in a simple and transparent way. We then generate new solvable potentials that describe interactions between electromagnetic field and matter, paying due attention to the subtleties inherent in the application of SUSY to higher dimensional problems. SUSY breaking problems related to the partner singularities are dicussed along with the paper.